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Using (10 087±44)×106 J/ψ events collected with the BESIII detector at a center-of-mass energy of s=3.097 GeV, the antihyperon-nucleon annihilation processes Λ¯p→K+π+π-+kπ0 (k=1, 2, 3) are studied at an incident Λ¯ momentum of approximately 1.074 GeV/c. The reactions Λ¯p→K+π+π-π0 and Λ¯p→K+π+π-2π0 are observed for the first time, with corresponding cross sections σΛ¯p→K+π+π-π0=8.5-1.1+1.2(stat)±0.4(syst) mb and σΛ¯p→K+π+π-2π0=7.9-1.7+1.9±0.4 mb. No significant signal is found for Λ¯p→K+π+π-3π0, and an upper limit of 7.2 mb is set at a 90% confidence level. Evidence for the K∗(892)+ resonance is seen in the K+π0 invariant mass spectrum MK+π0 for k=1, and the corresponding cross section for Λ¯p→K∗(892)+π+π- is measured to be σΛ¯p→K∗(892)+π+π-=12.5-3.4+3.8±1.2 mb. Owing to the limited statistics, possible interference effects are not considered. These findings offer crucial input to deepen our understanding of the antihyperon-nucleon interactions.